Problem 23 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(\alpha\) and \(\beta\) are imaginary cube roots of unity, then the value of \(\alpha^{4}+\beta^{28}+\frac{1}{\alpha \beta}\), is (a) 1 (b) \(-1\) (c) 0 (d) none of these

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Understand Cube Roots of Unity

Cube roots of unity are solutions to the equation \(x^3 = 1\). The roots are \(1, \omega, \omega^2\), where \(\omega = e^{2\pi i/3}\) and \(\omega^2 = e^{-2\pi i/3}\). Here, \(\alpha = \omega\) and \(\beta = \omega^2\).

Step 2: Simplify \(\alpha^4\) and \(\beta^{28}\)

Since \(\alpha = \omega\) and \(\beta = \omega^2\), and knowing \(\omega^3 = 1\), calculate \(\alpha^4 = \omega^4 = \omega\) and \(\beta^{28} = (\omega^2)^{28} = \omega^{56}\). Because \(\omega^{56} = \omega^{54+2} = (\omega^3)^{18}\omega^2 = 1^{18}\omega^2 = \omega^2\), we have \(\alpha^4 = \omega\) and \(\beta^{28} = \omega^2\).

Step 3: Calculate \(\frac{1}{\alpha \beta}\)

Since \(\alpha = \omega\) and \(\beta = \omega^2\), \(\alpha \beta = \omega \cdot \omega^2 = \omega^3 = 1\). Thus, \(\frac{1}{\alpha \beta} = 1\).

Step 4: Calculate \(\alpha^4 + \beta^{28} + \frac{1}{\alpha\beta}\)

Combine the results: \(\alpha^4 = \omega\), \(\beta^{28} = \omega^2\), and \(\frac{1}{\alpha \beta} = 1\). Therefore, the entire expression is \(\omega + \omega^2 + 1 = 0\), since the sum of the cube roots of unity is zero.

Key Concepts

Algebraic Identities

An algebraic identity is a mathematical statement that is always true for any values of the variables within a certain set. In solving algebraic problems, using identities simplifies computations and proofs. An essential identity when dealing with cube roots of unity is how they relate to the sum of roots. The equation \(x^3 = 1\) includes the roots \(1, \omega, \omega^2\).
One fundamental algebraic identity here is \(\omega^3 = 1\). This implies that applying powers of \(\omega\) repeatedly will eventually return you to the starting value, showing cyclical behavior. Another useful identity is \(\omega + \omega^2 + 1 = 0\), indicating that the complex cube roots of unity, along with the real cube root, sum to zero. This identity is crucial for simplifying expressions involving the cube roots of unity.
Understanding these identities helps in various algebraic manipulations and in simplifying complex problems as seen in the original exercise, where they were instrumental in reducing terms like \(\alpha^4 + \beta^{28} + \frac{1}{\alpha \beta}\). It illustrates how pre-established algebraic identities can transform complex equations into basic computations.

Complex Numbers

Complex numbers combine real and imaginary numbers into one system, allowing us to solve equations that do not have solutions within the realm of real numbers alone. A complex number is expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(i\) being the imaginary unit satisfying \(i^2 = -1\).
The concept of complex roots, such as the cube roots of unity, is built on this foundation. These roots are not part of the real numbers when \(xeq1\), hence their imaginary nature. In our context, the roots \(\omega = e^{2\pi i/3}\) and \(\omega^2 = e^{-2\pi i/3}\) are complex numbers:

• \(\omega\) represents one of the rotating positions on the complex plane, 120 degrees apart from \(1\).
• \(\omega^2\) is its complex conjugate, the second position in the triad, also 120 degrees apart from both \(1\) and \(\omega\).

Such complex numbers help us navigate problems concerning periodic phenomena and symmetry, similar to the periodic nature of trigonometric functions. By leveraging complex numbers, we can interpret these symmetries and cyclic patterns effectively.

Roots of Unity

Roots of unity are special numbers in complex number theory that represent regular divisions of the complex unit circle. The cube roots of unity specifically solve the equation \(x^3 = 1\), leading to the solutions \(1, \omega, \omega^2\). These roots hold a pivotal role in the study of cyclic phenomena, such as those encountered in wave mechanics and signal processing.

• The simplest root is \(1\), a real number.
• The other two roots \(\omega\) and \(\omega^2\) are complex, existing on the unit circle in the complex plane.

The property that the sum of the roots equals zero, \(\omega + \omega^2 + 1 = 0\), is especially powerful. This characteristic means that although individually, the roots circle back to 1 when cubed, collectively they exhibit balance, canceling each other out.
In the original exercise, identifying and simplifying \(\alpha = \omega\) and \(\beta = \omega^2\) played a key role. These roots manifest the complex interplay of cyclical symmetry, a natural pattern found in various branches of mathematics beyond just algebra.